Сегодня вода в реке выше, чем вчера.

Russian often omits the verb to be in the present tense.

So:

• Сегодня вода в реке выше, чем вчера.

literally works like:

• Today the water in the river higher than yesterday.

In natural English, you need is, but in Russian you usually do not say it in the present tense.

Compare:

• Вода холодная. = The water is cold.
• Он дома. = He is at home.

But in the past or future, forms of to be do appear:

• Вода была выше вчера. = The water was higher yesterday.
• Вода будет выше завтра. = The water will be higher tomorrow.

Выше is the comparative form, meaning higher.

• высокая = high, tall
• выше = higher

Because the sentence compares today with yesterday, Russian uses the comparative:

• Сегодня вода в реке выше, чем вчера.
• Today the water in the river is higher than yesterday.

If you said высокая, that would only describe the water as high, not compare it:

• Сегодня вода высокая sounds unnatural here, because the idea is specifically about a change in level.

So выше is the correct form for comparison.

Russian often expresses this idea as the water is higher, and that is completely natural.

So:

• вода в реке выше

means that the level of the water in the river has risen.

English often prefers:

• the river is higher
• the water level is higher

Russian can also say things like:

• Уровень воды в реке выше, чем вчера. = The water level in the river is higher than yesterday.

But вода в реке выше is normal and idiomatic.

Because в реке means in the river.

The noun река changes form after the preposition в:

• dictionary form: река
• after в for location: в реке

This is the prepositional case.

So:

• вода в реке = water in the river

You are not saying the river is the subject. The subject is вода = water.
в реке simply tells you where that water is.

It is the prepositional singular form of река.

The noun река is a feminine noun ending in -а.
For many nouns like this, the prepositional singular ends in -е:

• река → в реке
• книга → в книге
• школа → в школе

So in this sentence:

• в реке = in the river

The preposition в here answers where?, so it takes the prepositional case.

Here чем means than in a comparison.

So:

• выше, чем вчера = higher than yesterday

This is the standard way to make many comparisons in Russian:

• Он выше, чем я. = He is taller than me.
• Сегодня теплее, чем вчера. = Today is warmer than yesterday.
• Эта книга интереснее, чем та. = This book is more interesting than that one.

So in your sentence, чем introduces what today is being compared with: вчера.

Because Russian often compares directly with a time word like вчера if the meaning is clear.

• Сегодня ... выше, чем вчера.
  = Today ... is higher than yesterday.

This is natural and concise. Russian does not need to repeat the full idea like than it was yesterday.

English often fills in missing words mentally:

• higher than yesterday
  really means
• higher than it was yesterday

Russian does the same thing.

A fuller version is possible in some contexts, but it would usually sound heavier and less natural here.

Because both of these words can function as adverbs of time by themselves.

• сегодня = today
• вчера = yesterday

They do not need a preposition like on or in.

Examples:

• Сегодня холодно. = Today it is cold.
• Я видел его вчера. = I saw him yesterday.

So in your sentence:

• Сегодня sets the time of the statement.
• вчера is the point of comparison.

The sentence has a very normal, neutral word order:

• Сегодня | вода | в реке | выше, чем вчера.

This is roughly:

• Today | the water | in the river | is higher than yesterday.

Russian word order is more flexible than English, so other versions are possible, but the emphasis changes.

For example:

• Вода в реке сегодня выше, чем вчера.
  This puts water in the river first, then adds today.

• Выше сегодня вода в реке, чем вчера.
  Grammatically possible, but more marked and stylistic.

For a learner, the original sentence is a very good neutral pattern.

Here it is a comparative form used predicatively, and in practical learner terms you can think of it as higher.

It comes from высокий = high / tall.

Russian comparatives like выше, ниже, больше, меньше are often used in sentences without an extra verb:

• Дом выше. = The house is higher / taller.
• Температура ниже. = The temperature is lower.

So in your sentence, выше describes the state of the water: it is at a higher level.

You usually would not say that here.

Russian has two ways to form comparison:

1. Simple comparative

   • выше = higher

2. Analytical comparative with более

   • более высокий = more high

In everyday Russian, for common adjectives like высокий, the simple comparative is much more natural:

• вода выше = natural
• вода более высокая = possible in some formal or technical contexts, but awkward here

So the best version is:

• Сегодня вода в реке выше, чем вчера.

Yes. Russian sometimes uses the genitive case instead of чем after a comparative, especially with nouns or pronouns.

For example:

• Он выше меня. = He is taller than me.
• Этот дом выше того. = This house is taller than that one.

But in your sentence, the comparison is with вчера, which is an adverb of time, so чем is the natural choice:

• выше, чем вчера

You cannot really replace вчера with a genitive form the way you can with a noun like меня.

A very close breakdown is:

• Сегодня = today
• вода = water
• в реке = in the river
• выше = higher
• чем = than
• вчера = yesterday

So literally:

• Today water in the river higher than yesterday

Natural English:

• Today the water in the river is higher than yesterday.
• More idiomatic English: The water in the river is higher today than it was yesterday.

The stress is:

• сегОдня
• водА
• в рекЕ
• вЫше
• чем
• вчерА

A rough pronunciation guide:

• segODnya
• vaDA
• v riKYE
• VYshe
• chem
• fchera

A couple of useful notes:

• In connected speech, в реке often sounds close to вреке.
• чем is pronounced with a ch sound like in chess.

Russian has no articles like a, an, or the.

So:

• вода can mean water, the water
• река can mean a river, the river

The exact meaning comes from context.

In this sentence, English naturally uses the:

• the water in the river

But Russian simply says:

• вода в реке

with no article at all.